7.96/2.86	YES
7.96/2.87	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
7.96/2.87	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
7.96/2.87	
7.96/2.87	
7.96/2.87	Termination of the given RelTRS could be proven:
7.96/2.87	
7.96/2.87	(0) RelTRS
7.96/2.87	(1) RelTRSRRRProof [EQUIVALENT, 61 ms]
7.96/2.87	(2) RelTRS
7.96/2.87	(3) RelTRSRRRProof [EQUIVALENT, 20 ms]
7.96/2.87	(4) RelTRS
7.96/2.87	(5) RelTRSRRRProof [EQUIVALENT, 19 ms]
7.96/2.87	(6) RelTRS
7.96/2.87	(7) RelTRSRRRProof [EQUIVALENT, 12 ms]
7.96/2.87	(8) RelTRS
7.96/2.87	(9) RIsEmptyProof [EQUIVALENT, 0 ms]
7.96/2.87	(10) YES
7.96/2.87	
7.96/2.87	
7.96/2.87	----------------------------------------
7.96/2.87	
7.96/2.87	(0)
7.96/2.87	Obligation:
7.96/2.87	Relative term rewrite system:
7.96/2.87	The relative TRS consists of the following R rules:
7.96/2.87	
7.96/2.87	   f(g(x)) -> g(f(f(x)))
7.96/2.87	   f(h(x)) -> h(g(x))
7.96/2.87	   f'(s(x), y, y) -> f'(y, x, s(x))
7.96/2.87	
7.96/2.87	The relative TRS consists of the following S rules:
7.96/2.87	
7.96/2.87	   rand(x) -> x
7.96/2.87	   rand(x) -> rand(s(x))
7.96/2.87	
7.96/2.87	
7.96/2.87	----------------------------------------
7.96/2.87	
7.96/2.87	(1) RelTRSRRRProof (EQUIVALENT)
7.96/2.87	We used the following monotonic ordering for rule removal:
7.96/2.87	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(f(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(g(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(h(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(f'(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[3, 0], [0, 0]] 	* 	x_1 	 +  	[[2, 0], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [0, 0]] 	* 	x_3
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(rand(x_1)) =  	[[3], [3]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
7.96/2.87	Rules from R:
7.96/2.87	none
7.96/2.87	Rules from S:
7.96/2.87	
7.96/2.87	   rand(x) -> x
7.96/2.87	
7.96/2.87	
7.96/2.87	
7.96/2.87	
7.96/2.87	----------------------------------------
7.96/2.87	
7.96/2.87	(2)
7.96/2.87	Obligation:
7.96/2.87	Relative term rewrite system:
7.96/2.87	The relative TRS consists of the following R rules:
7.96/2.87	
7.96/2.87	   f(g(x)) -> g(f(f(x)))
7.96/2.87	   f(h(x)) -> h(g(x))
7.96/2.87	   f'(s(x), y, y) -> f'(y, x, s(x))
7.96/2.87	
7.96/2.87	The relative TRS consists of the following S rules:
7.96/2.87	
7.96/2.87	   rand(x) -> rand(s(x))
7.96/2.87	
7.96/2.87	
7.96/2.87	----------------------------------------
7.96/2.87	
7.96/2.87	(3) RelTRSRRRProof (EQUIVALENT)
7.96/2.87	We used the following monotonic ordering for rule removal:
7.96/2.87	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(f(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(g(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(h(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(f'(x_1, x_2, x_3)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [1, 0]] 	* 	x_3
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(s(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(rand(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
7.96/2.87	Rules from R:
7.96/2.87	
7.96/2.87	   f'(s(x), y, y) -> f'(y, x, s(x))
7.96/2.87	Rules from S:
7.96/2.87	none
7.96/2.87	
7.96/2.87	
7.96/2.87	
7.96/2.87	
7.96/2.87	----------------------------------------
7.96/2.87	
7.96/2.87	(4)
7.96/2.87	Obligation:
7.96/2.87	Relative term rewrite system:
7.96/2.87	The relative TRS consists of the following R rules:
7.96/2.87	
7.96/2.87	   f(g(x)) -> g(f(f(x)))
7.96/2.87	   f(h(x)) -> h(g(x))
7.96/2.87	
7.96/2.87	The relative TRS consists of the following S rules:
7.96/2.87	
7.96/2.87	   rand(x) -> rand(s(x))
7.96/2.87	
7.96/2.87	
7.96/2.87	----------------------------------------
7.96/2.87	
7.96/2.87	(5) RelTRSRRRProof (EQUIVALENT)
7.96/2.87	We used the following monotonic ordering for rule removal:
7.96/2.87	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(f(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(g(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(h(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(rand(x_1)) =  	[[3], [3]] 	 +  	[[3, 3], [3, 3]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
7.96/2.87	Rules from R:
7.96/2.87	
7.96/2.87	   f(h(x)) -> h(g(x))
7.96/2.87	Rules from S:
7.96/2.87	none
7.96/2.87	
7.96/2.87	
7.96/2.87	
7.96/2.87	
7.96/2.87	----------------------------------------
7.96/2.87	
7.96/2.87	(6)
7.96/2.87	Obligation:
7.96/2.87	Relative term rewrite system:
7.96/2.87	The relative TRS consists of the following R rules:
7.96/2.87	
7.96/2.87	   f(g(x)) -> g(f(f(x)))
7.96/2.87	
7.96/2.87	The relative TRS consists of the following S rules:
7.96/2.87	
7.96/2.87	   rand(x) -> rand(s(x))
7.96/2.87	
7.96/2.87	
7.96/2.87	----------------------------------------
7.96/2.87	
7.96/2.87	(7) RelTRSRRRProof (EQUIVALENT)
7.96/2.87	We used the following monotonic ordering for rule removal:
7.96/2.87	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(f(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(g(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(rand(x_1)) =  	[[3], [3]] 	 +  	[[3, 3], [3, 3]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	   <<<
7.96/2.87	 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
7.96/2.87	>>>
7.96/2.87	
7.96/2.87	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
7.96/2.87	Rules from R:
7.96/2.87	
7.96/2.87	   f(g(x)) -> g(f(f(x)))
7.96/2.87	Rules from S:
7.96/2.87	none
7.96/2.87	
7.96/2.87	
7.96/2.87	
7.96/2.87	
7.96/2.87	----------------------------------------
7.96/2.87	
7.96/2.87	(8)
7.96/2.87	Obligation:
7.96/2.87	Relative term rewrite system:
7.96/2.87	R is empty.
7.96/2.87	The relative TRS consists of the following S rules:
7.96/2.87	
7.96/2.87	   rand(x) -> rand(s(x))
7.96/2.87	
7.96/2.87	
7.96/2.87	----------------------------------------
7.96/2.87	
7.96/2.87	(9) RIsEmptyProof (EQUIVALENT)
7.96/2.87	The TRS R is empty. Hence, termination is trivially proven.
7.96/2.87	----------------------------------------
7.96/2.87	
7.96/2.87	(10)
7.96/2.87	YES
8.19/3.00	EOF